Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite lattices

Abstract

<p style='text-indent:20px;'>In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-<inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-<inline-formula><tex-math id="M2">\begin{document}$ {\mathcal{D}} $\end{document}</tex-math></inline-formula> attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.</p>

Invariant measures for first order nonlocal lattice dynamical systems with delays

“…Yang, Caraballo and Chen [49] constructed a family of invariant probability measures as well as periodic invariant probability measures and established some general results on the limiting behaviors of invariant measures and periodic invariant measures for nonautonomous dynamical systems. Invariant measures of lattice dynamical systems are also widely investigated by researchers; see e.g., [19,27,28,48,42,55,60], etc. Particularly, Zhao, Xue and Lukaszewicz [55] established the existence of invariant Borel probability measures for non-autonomous discrete Klein-Gordon-Schrödinger equations.…”